Psychology, 2010, 1, 252-260
doi:10.4236/psych.2010.14034 Published Online October 2010 (http://www.SciRP.org/journal/psych)
A Mathematical Study of the Dynamics of
Conscious Acquiring of Knowledge through
Reading and Cramming and the Process of Losing
Information from the Brain by
Natural Forgetting of Facts
Sudipto Roy1, Priyadarshi Majumdar2
1Department of Physics, Calcutta Institute of Engineering and Management, Kolkata, India; 2Jyotinagar Bidyasree Niketan H.S.
School, Kolkata, India.
Received May 29th, 2010; revised June 17th, 2010; accepted August 23rd, 2010.
ABSTRACT
We model the conscious learning process of human brain with a dynamical equation (cramming dynamics) by consid-
ering both the data entry and loss of data simultaneously. We show the analytical solution of the differential equation in
some special cases. We define some indexes like memory index, merit index, utilization index etc. Using them we can
measure the corresponding memory functions. Applications of this model have also been discussed. More general nu-
merical and analytical results are also presented at the end.
Keywords: Switching Function, Cramming Dynamics, Memory Index, Merit index, Utilization Index, Relative
Performance Index
1. Introduction
The memory retention mechanism in human brain is
quite a complicated process indeed. It is just like infor-
mation processing for a huge database. This database is
filled with all sorts of information that we use to go about
our everyday lives. The information is stored and re-
trieved as needed. No matter what we are doing, at any
instant or other, at anywhere, the memory is involved in
an active fashion. This complex system or network of
data processing is located in different parts of the brain
like the hippocampus and cortex. As these parts work in
tandem, memory begins to process and interacts with the
environment and its surroundings.
After understanding how we process memory, why do
we sometimes loose memories? It has been proven that
we forget simply because of a problem with encoding,
storage, retrieval, or a combination of any of these. The
first significant study in this field was carried out by
Ebbinghaus [1]. He studied the memorisation of non-
sense syllables. By repeatedly testing himself after va-
rious time periods and recording the results, he was the
first to describe the shape of the forgetting curve. On the
other hand acording to Eichenbaum [2], most forgetting
occurs very soon after learning. However, when mean-
ingful material is used, the forgetting curve is not so
steep. Memory also fades and become less reliable with
time and aging. An over flow of information may also
cause certain information to be forgotten as a result of
competition. Benfen ati [3] in his work examined the cel-
lular and molecular mechanisms that contribute to the
various forms of memory including short and long-term
memory, unconscious and conscious memory etc. Cro-
vitz et al. [4] suggested that a memory measurement (M)
can be expressed as a power function of time t as,
M
t



. Anderson [5] mentioned that the experimen-
tal power function curves are related to the mean taken
over all the subjects. Wixted [6] opposed the idea of
Anderson by showing that the power function also fits
better than the exponential function when data from the
individual subjects are fitted. Other important models of
the forgetting process are CHARM, due to Metcalfe [7],

A Mathematical Study of the Dynamics of Conscious Acquiring of Knowledge Through Reading and
Cramming and the Process of Losing Information from the Brain by Natural Forgetting of Facts
253
Chappell [8], Matrix Model, due to Humphreys [9] and
MINERVA II due to Hintzman [10]. The neural network
model predicted by Hopfield [11] was latter modified by
sikstrom [12] using the concept of bounded weights and
a distribution of learning rates. In a recent communica-
tion Stepanov [13] proposed a new model of memoriza-
tion dynamics using exponential functions.
In our work we have attempted to build up a physical
model of the memorizing process of the human brain (of
a learner in particular) undergoing a course of study with
a fixed duration of time. The act of learning is considered
here as a process of data storage in the brain. It is as-
sumed that one accumulates data while studying a sub-
ject consciously and there is a continuous process of data
loss, caused by several physiological and psychological
factors such as mental stress and fatigue etc. As this
memorizing and forgetting processes are continuous in
the said time interval hence we can express this process
as a dynamical equation to be explained in more details
in the section to follow. The dynamical equation inv olves
different parameters quantifying the capacity of the brain
from different aspects e.g. memorization ability, grasping
power etc. The solution to that differential equation of
learning process gives us a clear picture of how data are
being stored and lost continuously from the brain. We
have obtained the analytical solution of the said equation
in some particular limits and the numerical solution has
also been obtained as a result of system simulation using
MATLAB in Intel platform. The corresponding results
are mentioned in the sections to follow.
The idea behind choosing this particular model of the
brain and the basic assumptions are very simple and
based upon our common experiences. We have assumed
that the rate of data storage (accumulation rate) in the
brain at any instant can be calculated simply by subtract-
ing the rate of data loss from the rate of data entry. Ob-
viously the actual dynamics of the brain may not be ex-
actly the same as we have assumed but for all practical
purposes our work can mimic the activity of the br ain up
to a certain extent that will be cleared from our next
analysis. To support this claim we have shown that the
work of Ebbinghaus [1] will come as a special case of
our model. Also none of the previous models as men-
tioned earlier are as so much simple like ours and can
give a complete mathematical description of both the
learning and forgetting processes simultaneously using
the concept of switchin g process of the brain.
2. Mathematical Modeling
Let RS, RL and RE be the rate of data storage, the rate of
data loss and the rate of data entry in the brain at any
instant respectively. Let x(t) be the amount of data or
information already stored in the brain at any time t,
hence, the rate of storage at that moment is given by
SEL
dx
RRR
dt

 (1)
The rate of data entry can be enhanced by the factors
like intelligence, concentration, the ability of a person to
cope with the stressed situations etc. Experience tells us
that as we go on acquiring more and more knowledge
and thereby store more and more data, the rate of data
entry becomes slower due to some mental stress or brain
fatigue, as we generally perceive. As the accumulated
data increases in volume in the brain, the rate of data
entry must decrease. Hence, to give it a very simple
mathematical form, one can safely assume that at any
point of the learning process we have
() on
E
E
RStR. (2)
where ()St is a time dependent switching function (to
be discussed elaborately later on) whose value toggles
between 0 and 1 for it’s OFF and ON stages respectively.
During a conscious effort of cramming, this switch rem-
ains ON and otherwise it is OFF. on
E
R is actually the
rate of data entry for the ON state of the switch ()St .
Thus, during the ON state, we have on
EE
RR. During
the OFF state of ()St we have0
E
R. We define on
E
R
as Equation (3).
Here, C1 quantifies one’s intelligence, concentration,
eagerness, urgency of learning etc. Thus, it is a measure
of the traits of the learner that causes faster entry of data.
C denotes the maximum storage capacity, hence
x
C

.
The quantity /
x
C is the fraction of memory occupied
by information and therefore (1/ )
x
C is the fraction
of storage space still available for data entry. Common
experience tells us that, larger the va lue of /
x
C, greater
will be the difficulty in the further storage of data. The
parameter

is a positive quantity that may be called the
index of brain fatigue related to memorization (taking
care of the non-linearity). Since 1
x
C




is generally
a fraction,
E
on
R is smaller for larger values of

.
11
1
1
1/
on
E
C measure of grasping power, concentration, IQ and urgency of learningx
R C
measure of factorsimpedingaccumulation ofdataC
xC










(3)

A Mathematical Study of the Dynamics of Conscious Acquiring of Knowledge Through Reading and
Cramming and the Process of Losing Information from the Brain by Natural Forgetting of Facts
254
It is a common experience that one forgets information
more rapidly when the amount of accumulated data is
large. In mathematical terms, the larger the value of
/
x
C, greater will be the amount of data loss per unit
time. As the storage becomes higher the rate of loss bec-
omes more and more pronounced, possibly due to the
limitation of retention ab ility and the stress caused by the
load of already accumulated data. One may simply write
an expression for rate of data loss (RL), at any stage of
learning, as a function of the data (x) already stored in
the brain in the following way

2
β
L
x
/C
stress caused by data storage or data load
Rmemory retention abilityC 
(4)
The parameter C2 is a measure of one’s ability of
memory retention or memorizing ability. The term
(/ )
x
C

may be regarded as a measure of stress caused
by the accumulated data where the parameter

is int-
roduced to take care of the natural non-linearity of the
process. Like

, it is also a positive quantity. Since
(/ )
x
C is a fraction ,
L
R is smaller for higher values o f

. The parameter

may be called the stress endur-
ance index. A person with a larger value of

feels less
stressed by the accumulated data (i.e. less internal anxi-
ety about the necessity of retention of accumulated data).
With substitutions in (1), from (2, 3) and (4) we have
12
(/ )
() 1
dxxx C
StC
dtC C





 (5)
Let us define a dimensionless variable as/
X
xC

(with
X
varying from 0 to 1). Hence, in terms of X, (5)
can be expressed as

1
2
() 1 StC dX X
X
dt CCC


 (6)
3. Analysis
For any arbitrary learner the parameters C1 and C2 are
defined below
max
111 1
, where 01,CfC f (7)
max
222 2
, where 01,CfC f (8)
Here, max
1
C and max
2
Care the values of 1
C and
2
Cfor the best possible learner (ideal, being the most
intelligent, enthusiastic, diligen t and having the strongest
memory and greatest zeal for learning). Here, we define
the dimensionless parameters 12 and ff as the merit
index and the memory index respectively of any arbitrary
learner, quantifying one’s IQ and MRA (memory reten-
tion ability) respectively, relative to the best learner. For
the learner of the highest calibre or merit we have
12
1ff

. Using (7) and (8) in (6) we obtain

max
11 max
22
() 1 St fC dX X
X
dt CCf C



(9)
For convenience of calculation (without any loss of
generality) we may choose max max
12
1CC. If a system
of units can be defined for these quantities, they can al-
ways be chosen to satisfy this equality. Hence (9) can be
expressed as

1
2
() 1 St f dX X
X
dt CCf


 (10)
To determine the variation of X as a function of time, we
need to solve (10) for different functional forms of()St
(Figures 1-3). Where()St is a function determining the
duration for which the data entry channel remains open.
As long as one maintains a conscious learning effort,
()St remains 1, zero otherwise. During the time when
() 0St

we have from (10)
2
dX X
dt Cf

 (11)
Let m
T be the duration for which one maintains a
conscious memorizing effort without any break. Hence
we may write
()1 for 0m Stt T

 (12)
()0 for m Stt T

 (13)
The above criteria can very well be approximated by a
tan-hy perbolic function as gi ven below

1
()1tanh ()1
2m
Stt T 

 (14)
Figure 1. Variation of X with t for three different values of
C, with 12 0.6 0.6
f
=,
f
=, 6 10  , 200
m
T,

0.9α,

25β and S(t) follows (14).

A Mathematical Study of the Dynamics of Conscious Acquiring of Knowledge Through Reading and
Cramming and the Process of Losing Information from the Brain by Natural Forgetting of Facts
255
Figure 2. Variation of X with t for three different
β
, with
12
0.6, 0.6f=f= , 6 10ε=, 200
m
T= , =0.9α， =10C and
S(t) follows (14).
Figure 3. Variation of X with t for three different

, with
., .,
12
06 06ff ,6 10  200
m
T, 100C,

5

and
S(t) follows (14).
For a sufficiently large positive valu e of

, this func-
tion behaves almost exactly like (12, 13).
There may be another situation where the data entry
channel remains open intermittently. The learner can
alternat ely open and close th e channel in a period ic fash-
ion. Let T be the interval of time over which it remains
open and, for the next phase of the same duration it re-
mains closed. In this case, the plot of ()St vs. t should
have the pattern of a square wave, varying between 0 and
1. For numerical calculations in this case, ()St can be
approximated by a function of the following form (the
case is illustrated in Figure 4.
1
()tanh sin1
2m
t
St T





 



(15)
Figure 4. X-t variation with ,,
12
0.6 0.6ff 6 10ε,

5
m
T,,,

103 0.9Cβα and S(t) follows (15).
The sharpness of the square wave pattern of ()St in-
creases with higher values of

.
For a sufficiently large positive value of

, this function
behaves in a manner such that
()1 for 2(21),
0,1,2,3.......
mm
StnT tnT
n

 
 (16)
()0 for (21)(22),
0,1,2,3.......
mm
StnT tnT
n


 (17)
4. Analytical Solution for a Particular Case
For 1



, both growth and decay processes men-
tioned earlier are exponential in nature (as suggested by
(10, 11)), similar to that obtained by Ebbinghaus [1].
Under the boundary condition that, at 0 0,tXX, we
have from (10)
1212 0
12 12
2
12
exp ,
11
1
ff fft
XX
ff ff
Cf
With ff




 


 


(18)
It suggests that as t, 12
12
1
ff
Xff . Hence we
define 12
max 12
1
ff
Xff  and 12
max max12
1
Cf f
xCX ff 
,
as the saturation values of X and x respectively. For the
best learner, having12 1ff

, these quantities can be
expressed as max 1
2
b
X

and maxmax 2
bb
C
xCX . Here,
2
12
1
Cf
ff
  is the characteristic time constant determin-
ing the rapidity with which X reaches its saturation

A Mathematical Study of the Dynamics of Conscious Acquiring of Knowledge Through Reading and
Cramming and the Process of Losing Information from the Brain by Natural Forgetting of Facts
256
value max X. For smaller values of

, X varies at a faster
rate with time.
For 12 1ff, we have max
2
C
 . The duration
(m
T) of the interval, for which one continues a conscious
cramming effort, may be expressed as a multiple of
max

as max 2
m
NC
TN   . Hence.
2
12
2
(1 )
m
Tf
Nff
  (19)
Let us now define a dimensionless variable n such th at
m
tnT (20)
Combining (19) and (20) we obtain
12
2
(1 )
2
nNf f t
f 

 (21)
Incorporating all these results in (18) we get the foll-
owing expression of X representing the data storage
process up to the time of m tT we are getting

1
maxmax 0max
() exp
2
m
nNf
Xt TXXXX

 


(22)
While studying the behaviour of X beyond m tT

, we
need to solve (10) for 1

 and ()0St  under the
boundary condition that, at ,
mM
tTX X where
M
X
is the value of ()
m
X
tT at m
tT or equiva-
lently at 1n. The corresponding solution holds only
for m tT and is given by
2
() exp
m
mM
Tt
Xt TXCf


 

(23)
M
X
can be determined from (18) by using m tT

.
/2NC Then, by using (20) the above expression
takes the form of Equation (24).
It would be reasonable to express 0
X
as a fraction of
max
X for any learner. Taking into account the expres-
sion of max X we may have
12
0max 12
1
f
f
XX
f
f

 
, with 01

 (25)
Now using (24) and (25) we obtain

1
max max 2
(1 )
() 11expexp
22
m
Nf
N
n
Xt TXXf


 

 




 

(26)
The behavior of X, as a function of n, is obtained from
(22) and (26) for the ranges 01n and 1n resp-
ectively (see Figures 5-7).
5. Applications
Let us now consider the learning behaviour for a group
of students, preparing for a certain examination proc-
ess.The time allotted for preparation before the examina-
tion is 1 T. Let h
X
be the information gathered by the
best learner. Hence using 12 1ff, 1 tT

and
max



(/2C

) again.
1
0max
11 exp
22
h
T
XX



 




(27)
A student may make some delay while starting the
process of learning. Let 1 T

be the time utilized by any
arbitrary learner where 01


. The delay in starting
the learning process is1 (1 )T


. Let us define

as the
utilization index. For the most sincere student

= 1.
Let a
X
be the amount of knowledge acquired by any
arbitrary learner before the commencement of examina-
tion.
Thus fora
X
X

at 1 tT


with 2
12
1
Cf
ff
  we
have
12 121
0
12 12
exp
11
a
ff ffT
XX
ff ff




 


 

(28)
The performance of any arbitrary student in an exami-
nation, relative to the best learner, can be defined as
12 121
0
12 12
1
0max
exp
11
11 exp
22
a
r
h
ff ffT
X
ff ff
X
PXT
X








 

 






(29)
Let us call r P the relative performance index. We
can always express 1 T as a multiple of max

as
1max
2
C
T

 , with 0

 (30)
Assuming 00X

and substituting for 1 T and

we get see (Figures 8-13)
.
2
)1(
exp
2
)1(
exp
11
)(
22
21
0
21
21
21
21 































 f
nN
f
ffN
X
ff
ff
ff
ff
TtXm (24)

A Mathematical Study of the Dynamics of Conscious Acquiring of Knowledge Through Reading and
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257
12 1212
12 122
22 (1)
exp
11 2
1exp( )
a
r
h
f
fff ff
ff fff
X
PX


 





 
 
(31)
Defining 12
12
2
1
f
f
K
f
f
 we obtain
1
1exp
1-exp(- )
r
f
KK
P





 


 (32)
When the time available for study is sufficiently long,
for a sincere learner (with 1

), the exponential
terms in the last equation become negligible and it will
reduce to
12
12
2
1
r
f
f
PK
f
f

 (33)
Using experimental results one can determine C and
2
f
.
Let us suppose somebody performs two tests of
Figure 5. Graphical representation of (22), (23) and (24)
with parametric variations.
Figure 6. Graphical representation of (22), (23) and (24)
with parametric variations.
Figure 7. Graphical representation of (22), (23) and (24)
with parametric variations.
Figure 8. r
P
-1
f
variation for three different2
f
, drawn
on the basis of (31)
Figure 9. r
P
-2
f
variation for three different 1
f
, drawn
on the basis of (31).
Figure 10. r
P
–1
f
variation for three different

, drawn
on the basis of (31).

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Figure 11. r
P
-
μ
variation for three different η, drawn
on the basis of (31).
Figure 12. r
P
-
μ
variation for three different 1
f
,
drawn on the basis of (3 1).
Figure 13. r
P
-
μ
variation for three different 2
f
,
drawn on the basis of (3 1).
knowledge at tT and 2tT, resulting in T
X
X

and 2T
X
X respectively. One can determine the ratio
2/ TT
X
X as

22
ln //
TT
X
XTCf (34)
For the best learner (with21f

), let the corresponding
X’s be b
T
X
and 2
b
T
X
respectively. Hence


2
ln //
bb
TT
X
XTC (35)
While constructing (34) and (35) the underlying as-
sumption is that, the maximum data storage capacity C of
the brain does not vary from person to person. It is a
constant for all human beings in the adulthood. Again as



2
22
ln /
ln /
bb
TT
TT
X
X
f
X
X
 (36)
One can determine the memory index (2
f
) of any
learner using the grade points scored by him and by the
best learner also, in the two examinations held at tT

and 2tT

. Now from (35)

2
2ln ln
ln /bb
bb
TT
TT
TT
C
X
X
XX
  (37)
Using (33) and (36) we get



2
1
2
ln /
2ln/
rTT
bb
rTT
PXX
fPXX  (38)
One can determine the merit index1
f
of any individ-
ual from the grade points scored by him and the best
learner in two examination held at tT and 2tT

.
The validity of the above set of equations are ensured
if the data entry channel remains closed during the time
between tT

and 2tT

. It means that these equa-
tions hold good only if there is no conscious learning
effort during this interval of time, on the part of both
learners.
6. More Generalised Analysis
The common experience of learning tells us that if we
sustain the learning process (by keeping () 1St

) for an
indefinite period of time, the memory X will continue to
increase but with a gradually decreasing rate. The value
of X will asymptotically approach a definite saturation
level (max X), which is likely to depend on parameters
(12 ,,,,ffC

) of the model given in (10).
Therefore, at () 1St

and max XX, (10) is expe-
cted to show the following behavior

max 1max 2
10
X f
dX X
dt CCf



 (39)
Hence

1/ /
12max max
10ff XX
  (40)
The parameter 12
f
f is generally a fraction. Hence we
can write 12 1/ff


where 1  . Substituting this in
(40) we have

A Mathematical Study of the Dynamics of Conscious Acquiring of Knowledge Through Reading and
Cramming and the Process of Losing Information from the Brain by Natural Forgetting of Facts
259

1/ /
max max
10XX



 (41)
The real solutions of this polynomial are the intersec-
tions of the curve

1/
max yX


 with straight line
max
1yX . For max 1X, the curve will never have
0y as obtained for the straight line. Thus, we always
have max 1X which means that a learner will always
have
x
C. Theoretically, the highest possible value of
(/)
X
xC is 1. Here we have to consider the particular
real and positivemax
X (we definitely have some positive
roots as all the parameters are positive) which is less than
1. For an individual learner, it is always desirable to have
a saturation value (max X) that is as close to 1 as possible.
To ensure it, the curve


1/
max yX


  must be very
flat. Since max 01X, for a sufficiently large value of

we must have max 1X

 . Hence, for this large value
of

, one can increase the flatness of the curve by re-
ducing the value of

. From the definitions of

and

it is clear that, persons with larger values of the ratio
/

 must make better learning performances. Here we
see that larger values of this ratio increase the flatness of
the curve

1/
max yX


 , making max X closer to 1.
In Figure 14, we have a plot of max
X as a function of
/

 for three different values of

. Any of these
curves shows that as the ratio /

 increases, max X
increases. At any fixed value of /

 , the value of
max
X increases as

increases.
It is important to calculate the time required for reach-
ing the saturation level. We know that, as t, we
have max XX, where max X is the saturation value.
One can calculate the time required to reach a certain
fraction of max
X. Let us define optimum time (opt
T) as
the time required to reach 99% of the saturation value.
Putting () 1St  in (10) we find an analytical expres-
sion for opt T as
0.99
2
12
max
(1 )
X
opt
Xin
dX
TCf
f
fXX



 (42)
Here, in
X
indicates the initial amount of memory at t
= 0. (42) shows that the in tegrand diminishes for smaller
values of

and larger values of

since01X

.
Therefore, the ratio /

 may be expected to play a
significant role here. This integrand also decreases for
any increase in the value of the quantity 12
f
f. For a
definite set of numerical values of the model-parameters
we can estimate opt T numerically from (42) and can
compare it with m T For any learner it is desirable to
maximize max X and minimize opt T.
Let us consider a situation that allows one to continue
the conscious learning process up to the time of m tT

.
Figure 14. Variation of max
X as a function of β/α for
different values of
β
for C = 10, f1 = 0.6, f2 = 0.6, ε=
1000000, T = 200, X0 = 0, S(t) = 1.
Thus, fromm
tT

onwards we have ()0St

. Here,
m
T may be the time given for preparation before any
test of learning. Comparing m
T with opt
T, let us dis-
cuss different possible cases below.
a) mopt
TT: In this case the learner gets the percep-
tion of reaching the saturation level before having to stop
the learning process.
b) m opt
TT

: Here, the learner just reaches the satura-
tion level at the end of the time interval given for learn-
ing. It is worse compared to the last case.
c) m opt
TT

: This situation is the most undesirable
one. Here the learner has to stop the process before at-
taining the saturation level of learning.
To explor e the role played by the ratio /

 , we may
define average learning speed (av
S) as follows
max
0.99
av
opt
X
ST  (43)
In Figure 15, we have a plot of av S as a function of
the ratio /

 for three different values of

. Each
curve has an increasing trend with a gradually decreasing
slope. Although these curves have intersections with
each other at lower values of /

 , at higher values of
this ratio, av S becomes larger for greater values of

.
Let us now analyze the situation after one stop the learn-
ing process at m tT

. The system now follows the (11).
Since we have already discussed cases with 1


earlier,
we should now consider the cases with 1

. Consid-
ering the boundary condition that k
X
X at m tT

,
we may write
1
1
2
(1 )1
mk
TtX
XCf







 



(44)

A Mathematical Study of the Dynamics of Conscious Acquiring of Knowledge Through Reading and
Cramming and the Process of Losing Information from the Brain by Natural Forgetting of Facts
260
Figure 15. Variation of average learning speed (Sav) as a
function of β/α for different values of
β
for C = 10, f1
= 0.6, f2 = 0.6, ε= 1000000, T = 200, X0 = 0, S(t) = 1.
Here we have to take into account the physical reality
that, for m
tT we must always have 0X and
therefore 10X
 . (44) clearly shows that, 10X  
at any point of time beyond m tT

only if 1

. Thus,
the present study shows that, the model is logically ac-
ceptable only for 1

. Unlike exponential decay,
which was obtained for the case with 1

, we have the
following solution for 1


1/(1 )
1
2
(1 )1
mk
TtX
XCf








 
 




(45)
(45) suggests that X decreases with time but it remains
greater than zero (since m
tT and 1

). We never
lose our entire memory in the decay process.
Common experience tells us that when one is engaged
in a conscious learning process (i.e. () 1St ), we must
have /0dxdt . Then, from (10) we can write

1/
1,XX


  where 12 1/
f
f   (46)
The values of

and

should be such that (46) is
satisfied for the highest possible values of  and X.
7. Conclusions
We have proposed and analyzed a simple model of lea-
rning process. Some numerical results including simula-
tions are also presented. Learning and memorizing are
two most essential features of human brain. The model
can further be improved by considering more compli-
cated growth and decay process. These can be achieved
if we express the features of memorizing process as men-
tioned in (3) and (4), e.g. grasping power, stress, IQ etc.
separately in clear mathematical form with proper ex-
planations.
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